We have our own Parliament and pass all of our own laws. Under our constitutional arrangements with the UK, they remain responsible for our foreign affairs and defence.
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We fully control our own immigration policy and do not follow criteria set by the UK.
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UK, EU and EEA nationals are all permitted to enter Gibraltar upon production of a valid national identity card or passport issued by such a State. A family member is also admitted into Gibraltar on the same basis.  Such nationals are also permitted to reside in Gibraltar and cross the land frontier into Spain.
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US, Canadian, Australian and several South American nationals do not require a visa to enter Gibraltar. Other nationals can check visa entry requirement on the website of the Borders and Coast Guard Agency.
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We were a part of the EU but left with the UK following Brexit. The UK are in the process of negotiating a Brexit treaty with the EU in respect of Gibraltar.  This is expected to be finalised within the next weeks or months.  
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The principal aim of the Treaty is to allow Gibraltar residents free access to the Schengen zone via the land frontier with Spain. The frontier would, in effect, disappear and it is expected that Gibraltar residents, regardless of nationality, would have free movement throughout the whole of the Schengen zone. The details in this regard are expected to emerge in the next few weeks and months.